Theorem elmpps 35055

Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mppsval.p	⊢ 𝑃 = (mPreSt‘𝑇)
mppsval.j	⊢ 𝐽 = (mPPSt‘𝑇)
mppsval.c	⊢ 𝐶 = (mCls‘𝑇)

Assertion

Ref	Expression
elmpps	⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))

Proof of Theorem elmpps

Dummy variables 𝑎 𝑑 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ot 4630	. . 3 ⊢ ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
2		mppsval.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
3		mppsval.j	. . . 4 ⊢ 𝐽 = (mPPSt‘𝑇)
4		mppsval.c	. . . 4 ⊢ 𝐶 = (mCls‘𝑇)
5	2, 3, 4	mppsval 35054	. . 3 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}
6	1, 5	eleq12i 2818	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
7		oprabss 7508	. . . 4 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ ((V × V) × V)
8	7	sseli 3971	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
9	2	mpstssv 35021	. . . . . 6 ⊢ 𝑃 ⊆ ((V × V) × V)
10	9	sseli 3971	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
11	1, 10	eqeltrrid 2830	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
12	11	adantr 480	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)) → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
13		opelxp 5703	. . . 4 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
14		opelxp 5703	. . . . 5 ⊢ (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
15		simp1 1133	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → 𝑑 = 𝐷)
16		simp2 1134	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ℎ = 𝐻)
17		simp3 1135	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
18	15, 16, 17	oteq123d 4881	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ⟨𝑑, ℎ, 𝑎⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
19	18	eleq1d 2810	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ↔ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃))
20	15, 16	oveq12d 7420	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (𝑑𝐶ℎ) = (𝐷𝐶𝐻))
21	17, 20	eleq12d 2819	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (𝑎 ∈ (𝑑𝐶ℎ) ↔ 𝐴 ∈ (𝐷𝐶𝐻)))
22	19, 21	anbi12d 630	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ((⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)) ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
23	22	eloprabga 7509	. . . . . 6 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
24	23	3expa 1115	. . . . 5 ⊢ (((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
25	14, 24	sylanb 580	. . . 4 ⊢ ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
26	13, 25	sylbi 216	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
27	8, 12, 26	pm5.21nii 378	. 2 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))
28	6, 27	bitri 275	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3466  ⟨cop 4627  ⟨cotp 4629   × cxp 5665  ‘cfv 6534  (class class class)co 7402  {coprab 7403  mPreStcmpst 34955  mClscmcls 34959  mPPStcmpps 34960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-ot 4630  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpst 34975  df-mpps 34980

This theorem is referenced by:  mthmpps  35064  mclspps  35066